Adding Dimension : Is it Axiom?

Question

Let's suppose I had one dimension defined as $D_1 =\{x \mid x \in \Bbb R\}$

then I will add another dimension to make $D_2 = \{(x,y)\mid x,y \in \Bbb R\}$

then I will generalize this adding one more dimension process and call it $F$ which keep add up another dimension to previous n-dimension.

Here my question is,

Could this process F be defined? or this process is an axiom that we just accept it?

Answer 1

What you get is that $D_n = \mathbb R^n$ which is of dimension $n$. If we look at $\mathbb R^n$ as vector space, we can use basic facts from linear algebra:

1. Every vector space has a basis, i.e. linear independent set that spans whole vector space.
2. Any two bases of a vector space have same cardinality.

Cardinality, i.e. number of elements of basis is then called dimension. In the case of $\mathbb R^n$ it is $n$, as expected, because we can see that set $\{(1,0,\ldots,0),(0,1,\ldots,0),\ldots,(0,0,\ldots,1)\}$ is linearly independent and spans the whole space.

If you want to build $n+1$-dimensional vector space from $n$-dimensional, you can use that $\mathbb R^{n+1}\cong \mathbb R^n\times \mathbb R$.

Answer 2

The operation can be defined, and is usually called "taking the direct product with $\Bbb R$", with the notation $D_n\times \Bbb R=D_{n+1}$ (some purists might object to me using $=$ here, but for now that's not something you need to worry about).

Exactly how it's defined is dependent on what context you're in (are these sets, topological spaces, metric spaces, vector spaces, groups, rings, or something else?), but intuitively they all correspond to exactly what you do here: simply tacking on another coordinate at the end.